Evaluate the line integral of the velocity u=√√√y, v=x-y over the following paths (a) x, y=t (b) x=2, y=t where 01. (c) x=1, y=12 (d) x, y = (³
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- 3.1. The velocity at a point in a fluid for a one-dimensional flow may be given in the Eulerian coordinates by u == AxBt. Show that x = f(x, t) in the Lagrange coordinates can be obtained from the Eulerian system. The in- itial position of the fluid particle is designated by x) and the initial time to = 0 may be assumed.1) Particle Motion and Coordinate Systems y y = 0.001x² %3D 100 m The jet pictured above is following the given curve. All of the question below are related to the moment pictured a) If the jet's altitude is increasing at a rate of 40 m/s, find its velocity vector (in Cartesian) b) Find the speed of the jet c) In addition, if the rate of change of the jet's vertical speed is increasing at 15 m/,2, find the acceleration vector d) Find the tangential component of the acceleration vector, v e) Find the normal component of the acceleration vector f) Find the radius of curvature of the jet's path using the information from parts b) and e)4. The velocity vectors of three flow fileds are given as V, = axĩ + bx(1+1)}+ tk , V, = axyi + bx(1+t)j , and V3 = axyi – bzy(1+t)k where coefficients a and b have constant values. Is it correct to say that flow field 1 is one-, flow filed 2 is two-, and flow filed 3 is three-dimensional? Are these flow fields steady or unsteady?
- If a, = 100 rad/s CCW, find the velocity of point E using the instant center method. Show the velocity vector VE, on the figure. A 02 2 70° E AB = 1.0" BC = 1.75" CD = 2.0" DE = 0.8" AD = 3.0" B 4 %3D 3.Find the Divergence (Divf) and curl(f) for the following function S(x, y,z) = xyz (xi + yj + zk).(3a)The first time derivative of vector A cross vector B is equal to the first time derivative of vector B cross vector A. The first time derivative of vector A cross vector B is equal to the first time derivative of vector B cross vector A. True False (b) Suppose you're interested in the first time-derivative of velocity vector v(t) = (5t2)i + (2t)j. Which of the following expressions represents the first time-derivative of two times v(t)? Suppose you're interested in the first time-derivative of velocity vector v(t) = (5t2)i + (2t)j. Which of the following expressions represents the first time-derivative of two times v(t)? (10t)i + (2)j (20t)i + (4)j (10)i (5/3)(t3)i + (t2)j
- Evaluate the integral by first modifying the form of the integrand and then making an appropriate substitution, if needed. 8t +9 dt = +Cy x = r cos 0 V = Or y = r sine r = √x² + y² χ Flow in "solid body rotation" acts like a solid spinning around an axis. The streamlines are circular, the velocity is purely tangential, and the velocity magnitude is V = r, where is the angular velocity (positive counter-clockwise) and r is the radius. (a) Express the velocity vector V as a function of x and y. (b) Calculate the curl of the velocity vector V × V, indicating clearly the direction of the resulting vector.1- The "divergence grad 6" can be written as: a: V(V.) b: Vx(V) e: V.(V.) d: V.(V) 2-The acceleration vector in plane polar coordinates is a = (-ro?)e, +(rö + 2ró)eo If a particle moves on a circle path with constant radius b then the radial and transverse components respectively are: a: be and bö b: b and 2bė e: B - bộ and zero d: zero and b- bo? 3- The concept of impulse can be obtained if the force is: a: F(x) b: F(t) e: F(v) d: constant 4- The position vector of a particle can be written in cylindrical coordinates (R, . z) as: a: r=ReR b: r-R eg +o e, + Ze, e: r=R eg +Ze, d: r=ReR + e,
- When a valve is opened, a certain fluid flows through the choke duct or valve (see figure), according to the relationship: V= V (1 + x/L) i Determine a) If the flow is stationary or transient. b) The acceleration (ax) of the fluid applying Euler's approach. c) The position of the particle as a function of time at x = 0 and t = 0. d) Determine the acceleration of the particle as a function of time.(d) Given a velocity graph as shown in Figure Q.2(d). Construct the a – t and s – t graphs. ERSIT v (m/s) v = 0.6t3/ t(s) 15 25 Figure Q.2(d)Ex.5.3 The position of a particle moving along the x-axis is described by x = t3 – 108t in. where t is the time in sec. For the time interval t = 0 to t = 10 s (a) plot the position, velocity and acceleration as a function of time (b) find the displacement of the particle and (c) determine the distance traveled by the particle. -